Convolution Idempotents with a given Zero-set

Abstract

We investigate the structure of N-length discrete signals h satisfying h ∗ h = h that vanish on a given set of indices. We motivate this problem from examples in sampling, Fuglede’s conjecture, and orthogonal interpolation of bandlimited signals. When N = p M is a prime power, we characterize all such h with a prescribed zero set in terms of basep expansions of nonzero indices in F − 1 h